The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence...The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence of exponential attractor are proved. The upper bounds of its fractal dimension are also estimated.展开更多
Abstract In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the gl...Abstract In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the global attractor are derived.展开更多
This paper deals with the asymptotic behaviour of solutions for thegeneralized symmetric regularized long wave equations with dissipation term. We first show theexistence of global weak attractors for the periodic ini...This paper deals with the asymptotic behaviour of solutions for thegeneralized symmetric regularized long wave equations with dissipation term. We first show theexistence of global weak attractors for the periodic initial value problem of this equations in H^2x H^1. And then by an energy equation and an idea of Ghidaglia and Guo, we conclude that the globalweak attractor is actually the global strong attractor for S(t) in H^2 (Ω) x H^1 (Ω). The finitedimensionality of the global attractor is also established.展开更多
This paper presents a meshless method for the nonlinear generalized regularized long wave （GRLW） equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtain...This paper presents a meshless method for the nonlinear generalized regularized long wave （GRLW） equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method. A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm. Compared with numerical methods based on mesh, the meshless method for the GRLW equation only requires the.scattered nodes instead of meshing the domain of the problem. Some examples, such as the propagation of single soliton and the interaction of two solitary waves, are given to show the effectiveness of the meshless method.展开更多
The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the ...The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.展开更多
This paper investigates the solitary wave solutions of the (2+1)-dimensional regularized long-wave (2DRLG) equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in pl...This paper investigates the solitary wave solutions of the (2+1)-dimensional regularized long-wave (2DRLG) equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas and (2+1) dimensional Davey-Stewartson (DS) equation which is governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multidimensional lattice. By using extended mapping method technique, we have shown that the 2DRLG-2DDS equations can be reduced to the elliptic-like equation. Then, the extended mapping method is used to obtain a series of solutions including the single and the combined non degenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear partial differential equations (NLPDEs).展开更多
By using the homogeneous balance principle（HBP）, we derive a Backtund transformation（BT） to the generalized dispersive long wave equation with variable coefficients.Based on the BT, we give many kinds of the exact...By using the homogeneous balance principle（HBP）, we derive a Backtund transformation（BT） to the generalized dispersive long wave equation with variable coefficients.Based on the BT, we give many kinds of the exact solutions of the equation, such as, singlesolitary solutions, multi-soliton solutions and generalized exact solutions.展开更多
The element-free Galerkin （EFG） method is used in this paper to find the numerical solution to a regularized long-wave （RLW） equation. The Galerkin weak form is adopted to obtain the discrete equations, and the es...The element-free Galerkin （EFG） method is used in this paper to find the numerical solution to a regularized long-wave （RLW） equation. The Galerkin weak form is adopted to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. The effectiveness of the EFG method of solving the RLW equation is investigated by two numerical examples in this paper.展开更多
In this article, we study the Lax pairs of -dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formula...In this article, we study the Lax pairs of -dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.展开更多
The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics,physics,and engineering disciplines.This article intends to analyze several traveling wave solutions f...The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics,physics,and engineering disciplines.This article intends to analyze several traveling wave solutions for themodified regularized long-wave(MRLW)equation using several approaches,namely,the generalized algebraic method,the Jacobian elliptic functions technique,and the improved Q-expansion strategy.We successfully obtain analytical solutions consisting of rational,trigonometric,and hyperbolic structures.The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation.The adaptive moving mesh method evenly distributes the points on the high error areas.This method perfectly and strongly reduces the error.We compare the constructed exact and numerical results to ensure the reliability and validity of the methods used.To better understand the considered equation’s physical meaning,we present some 2D and 3D figures.The exact and numerical approaches are efficient,powerful,and versatile for establishing novel bright,dark,bell-kink-type,and periodic traveling wave solutions for nonlinear PDEs.展开更多
The present study considers wave scattering phenomena around a cylindrical island mounted on a general axisymmetric topography or a general submerged truncated axi-symmetric shoal based on the mild-slope equation. The...The present study considers wave scattering phenomena around a cylindrical island mounted on a general axisymmetric topography or a general submerged truncated axi-symmetric shoal based on the mild-slope equation. The method of separation of variables and Taylor series expansion are invoked to find the approximate solution to the variable water depth region which varies proportionally to an arbitrary power of radial distance. Validations against the solutions for the combined wave refraction and diffraction around a cylindrical island mounted on a paraboloidal shoal of Liu et al. in 2004 and the scattering and trapping of wave energy by a submerged truncated paraboloidal shoal of Lin and Liu in 2007 show excellent agreements as the power of radial distance being equal to two. For the solutions of wave refraction and diffraction around a cylindrical island mounted on a shoal with depth proportionally to an arbitrary power of radial distance, good agreements with Zhai et al.’s(2013) solutions are demonstrated. Since the robustness of the assumption of a general axi-symmetric geometry based on an arbitrary power variability of the radial distance, the present solution can be very conveniently employed to investigate the effects of bottom topography on wave scattering and trapping patterns.展开更多
An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the...An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.展开更多
In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Backlund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present...In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Backlund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations.展开更多
The existence and uniqueness of the global smooth solution to the initial-boundary valueproblem of a system of multi-dimensions SRWE are proved. The sufficient conditions of 'blowingup' of the solution are given.
The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq-Burger and approximate long water wave equations by using the generalized Kudryashov method.The fractional differential...The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq-Burger and approximate long water wave equations by using the generalized Kudryashov method.The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann-Liouville derivative sense.Applying the generalized Kudryashov method through with symbolic computer maple package,numerous new exact solutions are successfully obtained.All calculations in this study have been established and verified back with the aid of the Maple package program.The executed method is powerful,effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.展开更多
The author demonstrate that the two-point boundary value problemhas a solution (A,P(8)), where III is the smallest parameter, under the minimal stringent resstrictions oil f(8), by applying the shooting and regularisa...The author demonstrate that the two-point boundary value problemhas a solution (A,P(8)), where III is the smallest parameter, under the minimal stringent resstrictions oil f(8), by applying the shooting and regularisation methods. In a classic paper)Kolmogorov et. al. studied in 1937 a problem which can be converted into a special case of theabove problem.The author also use the solutioll (A, p(8)) to construct a weak travelling wave front solutionu(x, t) = y((), (= x -- Ct, C = AN/(N + 1), of the generalized diffusion equation with reactionO { 1 O.IN￣1 OUI onde L k(u) i ox: &) ￣ & = g(u),where N > 0, k(8) > 0 a.e. on [0, 1], and f(s):= ac i: g(t)kl/N(t)dt is absolutely continuouson [0, 11, while y(() is increasing and absolutely continuous on (--co, +co) and(k(y(())ly,(OI'), = g(y(()) -- Cy'(f) a.e. on (--co, +co),y(￣oo) ￣ 0, y(+oo) ￣ 1.展开更多
基金ProjectsupportedbytheNationalNaturalScienceFoundationofChina (No .1 0 2 71 0 3 4)
文摘The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence of exponential attractor are proved. The upper bounds of its fractal dimension are also estimated.
文摘Abstract In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the global attractor are derived.
基金This research is supported by the National Natural Science Foundation of China(Grant 10271034).
文摘This paper deals with the asymptotic behaviour of solutions for thegeneralized symmetric regularized long wave equations with dissipation term. We first show theexistence of global weak attractors for the periodic initial value problem of this equations in H^2x H^1. And then by an energy equation and an idea of Ghidaglia and Guo, we conclude that the globalweak attractor is actually the global strong attractor for S(t) in H^2 (Ω) x H^1 (Ω). The finitedimensionality of the global attractor is also established.
基金supported by the National Natural Science Foundation of China (Grant No. 10871124)the Innovation Program of the Shanghai Municipal Education Commission,China (Grant No. 09ZZ99)
文摘This paper presents a meshless method for the nonlinear generalized regularized long wave （GRLW） equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method. A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm. Compared with numerical methods based on mesh, the meshless method for the GRLW equation only requires the.scattered nodes instead of meshing the domain of the problem. Some examples, such as the propagation of single soliton and the interaction of two solitary waves, are given to show the effectiveness of the meshless method.
基金supported by the National Natural Science Fundation of China (No. 11061021)the Science Research of Inner Mongolia Advanced Education (Nos. NJ10006, NJ10016, and NJZZ12011)the National Science Foundation of Inner Mongolia (Nos. 2011BS0102 and 2012MS0106)
文摘The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.
文摘This paper investigates the solitary wave solutions of the (2+1)-dimensional regularized long-wave (2DRLG) equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas and (2+1) dimensional Davey-Stewartson (DS) equation which is governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multidimensional lattice. By using extended mapping method technique, we have shown that the 2DRLG-2DDS equations can be reduced to the elliptic-like equation. Then, the extended mapping method is used to obtain a series of solutions including the single and the combined non degenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear partial differential equations (NLPDEs).
基金Supported by the Natural Science Foundation of Education Committee of Henan Province（2003110003）
文摘By using the homogeneous balance principle（HBP）, we derive a Backtund transformation（BT） to the generalized dispersive long wave equation with variable coefficients.Based on the BT, we give many kinds of the exact solutions of the equation, such as, singlesolitary solutions, multi-soliton solutions and generalized exact solutions.
基金supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y6110007)
文摘The element-free Galerkin （EFG） method is used in this paper to find the numerical solution to a regularized long-wave （RLW） equation. The Galerkin weak form is adopted to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. The effectiveness of the EFG method of solving the RLW equation is investigated by two numerical examples in this paper.
基金The project supported by National Natural Science Foundation of China under Grant No.10101025
文摘In this article, we study the Lax pairs of -dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.
文摘The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics,physics,and engineering disciplines.This article intends to analyze several traveling wave solutions for themodified regularized long-wave(MRLW)equation using several approaches,namely,the generalized algebraic method,the Jacobian elliptic functions technique,and the improved Q-expansion strategy.We successfully obtain analytical solutions consisting of rational,trigonometric,and hyperbolic structures.The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation.The adaptive moving mesh method evenly distributes the points on the high error areas.This method perfectly and strongly reduces the error.We compare the constructed exact and numerical results to ensure the reliability and validity of the methods used.To better understand the considered equation’s physical meaning,we present some 2D and 3D figures.The exact and numerical approaches are efficient,powerful,and versatile for establishing novel bright,dark,bell-kink-type,and periodic traveling wave solutions for nonlinear PDEs.
基金financially supported by the Ministry of Science and Technology of Taiwan,China(Grant No.MOST 107-2221-E-992)
文摘The present study considers wave scattering phenomena around a cylindrical island mounted on a general axisymmetric topography or a general submerged truncated axi-symmetric shoal based on the mild-slope equation. The method of separation of variables and Taylor series expansion are invoked to find the approximate solution to the variable water depth region which varies proportionally to an arbitrary power of radial distance. Validations against the solutions for the combined wave refraction and diffraction around a cylindrical island mounted on a paraboloidal shoal of Liu et al. in 2004 and the scattering and trapping of wave energy by a submerged truncated paraboloidal shoal of Lin and Liu in 2007 show excellent agreements as the power of radial distance being equal to two. For the solutions of wave refraction and diffraction around a cylindrical island mounted on a shoal with depth proportionally to an arbitrary power of radial distance, good agreements with Zhai et al.’s(2013) solutions are demonstrated. Since the robustness of the assumption of a general axi-symmetric geometry based on an arbitrary power variability of the radial distance, the present solution can be very conveniently employed to investigate the effects of bottom topography on wave scattering and trapping patterns.
文摘An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.
基金Acknowledgments This work is supported by the National Natural Science Foundation of China （Grant No. 11462019） and the Scientific Research Foundation of Inner Mongolia University for Nationalities （Grant No. NMD1306）. The author would like to thank the referees for their time and comments.
文摘In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Backlund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations.
文摘The existence and uniqueness of the global smooth solution to the initial-boundary valueproblem of a system of multi-dimensions SRWE are proved. The sufficient conditions of 'blowingup' of the solution are given.
文摘The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq-Burger and approximate long water wave equations by using the generalized Kudryashov method.The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann-Liouville derivative sense.Applying the generalized Kudryashov method through with symbolic computer maple package,numerous new exact solutions are successfully obtained.All calculations in this study have been established and verified back with the aid of the Maple package program.The executed method is powerful,effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.
文摘The author demonstrate that the two-point boundary value problemhas a solution (A,P(8)), where III is the smallest parameter, under the minimal stringent resstrictions oil f(8), by applying the shooting and regularisation methods. In a classic paper)Kolmogorov et. al. studied in 1937 a problem which can be converted into a special case of theabove problem.The author also use the solutioll (A, p(8)) to construct a weak travelling wave front solutionu(x, t) = y((), (= x -- Ct, C = AN/(N + 1), of the generalized diffusion equation with reactionO { 1 O.IN￣1 OUI onde L k(u) i ox: &) ￣ & = g(u),where N > 0, k(8) > 0 a.e. on [0, 1], and f(s):= ac i: g(t)kl/N(t)dt is absolutely continuouson [0, 11, while y(() is increasing and absolutely continuous on (--co, +co) and(k(y(())ly,(OI'), = g(y(()) -- Cy'(f) a.e. on (--co, +co),y(￣oo) ￣ 0, y(+oo) ￣ 1.
